Subgroups — C₂×C₁₃×#2615 #15720944

Order 3328.

6166 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×#121	C₂	1	3	yes	maximal, complemented
3	C₁₃×#2615	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₃×#393	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#392	C₂	1	6	yes	maximal, complemented
6	C₂×C₁₃×#380	C₂	1	3	yes	maximal, complemented
7	C₂²×C₁₃×#41	—	2	24	no	—
8	C₂²×C₁₃×#41	C₂²	1	8	yes	complemented
9	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₁₃	C₂²	1	3	yes	—
11	C₂³×D₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₁₃	C₂²	1	6	yes	—
13	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
14	C₂×C₁₃×#121	—	2	24	no	—
15	C₂×C₁₃×#121	C₂²	1	24	yes	complemented
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
17	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	6	yes	—
18	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	6	yes	—
19	C₁₃×#393	C₂²	1	16	yes	complemented
20	C₁₃×#392	C₂²	1	48	yes	complemented
21	C₁₃×#380	C₂²	1	24	yes	complemented
22	C₂×C₁₃×#95	C₂²	1	6	yes	—
23	C₂⁵×C₁₃	C₂³	1	2	yes	—
24	C₂⁵×C₁₃	C₂³	1	3	yes	—
25	C₂×C₁₃×#41	—	2	128	no	—
26	C₂×C₁₃×#41	—	2	96	no	—
27	C₂²×Q₈×C₁₃	—	2	12	no	—
28	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
29	C₂²×D₄×C₁₃	—	2	24	no	—
30	C₂²×D₄×C₁₃	—	2	24	no	—
31	C₂²×D₄×C₁₃	—	2	12	no	—
32	C₂²×D₄×C₁₃	—	2	12	no	—
33	C₂²×D₄×C₁₃	—	4	24	no	—
34	C₂²×D₄×C₁₃	C₂³	1	12	yes	—
35	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
36	C₂²×D₄×C₁₃	C₂³	1	24	yes	—
37	C₂³×C₄×C₁₃	—	2	24	no	—
38	C₂³×C₄×C₁₃	D₄	1	24	yes	complemented
39	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
40	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
41	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
42	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#121	—	2	96	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
46	C₂×C₁₃×C₂²⋊C₄	—	2	24	no	—
47	C₂×C₁₃×C₂²⋊C₄	—	4	12	no	—
48	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
49	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
50	C₁₃×#95	C₂³	1	24	yes	—
51	C₂×#2615	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	2	12	no	—
53	C₂⁴×C₁₃	—	2	6	no	—
54	C₂⁴×C₁₃	—	2	12	no	—
55	C₂⁴×C₁₃	—	2	6	no	—
56	C₂⁴×C₁₃	—	2	12	no	—
57	C₂⁴×C₁₃	—	2	12	no	—
58	C₂⁴×C₁₃	—	2	4	no	—
59	C₂⁴×C₁₃	C₂×D₄	1	12	yes	—
60	C₂⁴×C₁₃	C₂⁴	1	4	yes	—
61	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
62	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₁₃×#41	—	2	256	no	—
64	C₂×Q₈×C₁₃	—	2	32	no	—
65	C₂×Q₈×C₁₃	—	2	24	no	—
66	C₂×D₄×C₁₃	—	2	96	no	—
67	C₂×D₄×C₁₃	—	2	48	no	—
68	C₂×D₄×C₁₃	—	2	48	no	—
69	C₂×D₄×C₁₃	—	2	24	no	—
70	C₂×D₄×C₁₃	—	2	24	no	—
71	C₂×D₄×C₁₃	—	4	48	no	—
72	C₂×D₄×C₁₃	—	4	96	no	—
73	C₂×D₄×C₁₃	—	4	16	no	—
74	C₂²×C₄×C₁₃	—	2	12	no	—
75	C₂²×C₄×C₁₃	—	2	96	no	—
76	C₂²×C₄×C₁₃	—	2	48	no	—
77	C₂²×C₄×C₁₃	—	2	12	no	—
78	C₂²×C₄×C₁₃	—	2	24	no	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	48	yes	complemented
82	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
84	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
85	C₁₃×C₄⋊C₄	—	2	24	no	—
86	C₁₃×C₂²⋊C₄	—	2	48	no	—
87	C₁₃×C₂²⋊C₄	—	4	24	no	—
88	C₂³×#41	C₂×C₁₃	1	1	yes	characteristic, complemented
89	C₂²×#121	C₂×C₁₃	1	3	yes	complemented
90	#2615	C₂×C₁₃	1	16	yes	direct product factor
91	C₂×#393	C₂×C₁₃	1	2	yes	complemented
92	C₂×#392	C₂×C₁₃	1	6	yes	complemented
93	C₂×#380	C₂×C₁₃	1	3	yes	complemented
94	C₂³×C₁₃	—	2	24	no	—
95	C₂³×C₁₃	—	2	24	no	—
96	C₂³×C₁₃	—	2	24	no	—
97	C₂³×C₁₃	—	2	12	no	—
98	C₂³×C₁₃	—	2	6	no	—
99	C₂³×C₁₃	—	2	12	no	—
100	C₂³×C₁₃	—	2	6	no	—
101	C₂³×C₁₃	—	2	12	no	—
102	C₂³×C₁₃	—	2	12	no	—
103	C₂³×C₁₃	—	2	4	no	—
104	C₂³×C₁₃	—	4	24	no	—
105	C₂³×C₁₃	—	4	24	no	—
106	C₂³×C₁₃	—	4	8	no	—
107	C₂³×C₁₃	—	4	24	no	—
108	C₂³×C₁₃	—	4	24	no	—
109	C₂³×C₁₃	C₂²×D₄	1	12	yes	—
110	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
111	C₂³×C₁₃	H(5,𝔽₂)	1	6	yes	—
112	C₂³×C₁₃	2₋¹⁺⁴	1	2	yes	—
113	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
114	Q₈×C₁₃	—	2	32	no	—
115	D₄×C₁₃	—	2	96	no	—
116	D₄×C₁₃	—	4	96	no	—
117	D₄×C₁₃	—	4	16	no	—
118	C₂×C₄×C₁₃	—	2	24	no	—
119	C₂×C₄×C₁₃	—	2	12	no	—
120	C₂×C₄×C₁₃	—	2	96	no	—
121	C₂×C₄×C₁₃	—	2	12	no	—
122	C₂×C₄×C₁₃	—	2	24	no	—
123	C₂×C₄×C₁₃	—	4	24	no	—
124	C₂×C₄×C₁₃	#120	1	16	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
127	C₂²×#41	—	2	24	no	—
128	C₂²×#41	C₂²×C₁₃	1	8	yes	complemented
129	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
130	C₂³×D₄	C₂²×C₁₃	1	3	yes	—
131	C₂³×D₄	C₂²×C₁₃	1	1	yes	characteristic
132	C₂³×D₄	C₂²×C₁₃	1	6	yes	—
133	C₂⁴×C₄	C₂²×C₁₃	1	3	yes	—
134	C₂×#121	—	2	24	no	—
135	C₂×#121	C₂²×C₁₃	1	24	yes	complemented
136	C₂²×C₄⋊C₄	C₂²×C₁₃	1	3	yes	—
137	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	6	yes	—
138	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	6	yes	—
139	#393	C₂²×C₁₃	1	16	yes	complemented
140	#392	C₂²×C₁₃	1	48	yes	complemented
141	#380	C₂²×C₁₃	1	24	yes	complemented
142	C₂×#95	C₂²×C₁₃	1	6	yes	—
143	C₂²×C₁₃	—	2	12	no	—
144	C₂²×C₁₃	—	2	12	no	—
145	C₂²×C₁₃	—	2	12	no	—
146	C₂²×C₁₃	—	4	48	no	—
147	C₂²×C₁₃	—	4	12	no	—
148	C₂²×C₁₃	—	4	12	no	—
149	C₂²×C₁₃	—	4	4	no	—
150	C₂²×C₁₃	—	4	12	no	—
151	C₂²×C₁₃	—	4	12	no	—
152	C₂²×C₁₃	—	8	16	no	—
153	C₂²×C₁₃	C₂×#120	1	4	yes	—
154	C₂²×C₁₃	#535	1	12	yes	—
155	C₂²×C₁₃	#537	1	12	yes	—
156	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
157	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	3	yes	—
158	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
159	C₄×C₁₃	—	2	12	no	—
160	C₄×C₁₃	—	4	12	no	—
161	C₄×C₁₃	C₂×#120	1	8	yes	—
162	C₂⁵	C₂³×C₁₃	1	2	yes	—
163	C₂⁵	C₂³×C₁₃	1	3	yes	—
164	C₂×#41	—	2	128	no	—
165	C₂×#41	—	2	96	no	—
166	C₂²×Q₈	—	2	12	no	—
167	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
168	C₂²×D₄	—	2	24	no	—
169	C₂²×D₄	—	2	24	no	—
170	C₂²×D₄	—	2	12	no	—
171	C₂²×D₄	—	2	12	no	—
172	C₂²×D₄	—	4	24	no	—
173	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
174	C₂²×D₄	C₂³×C₁₃	1	12	yes	—
175	C₂²×D₄	C₂³×C₁₃	1	24	yes	—
176	C₂³×C₄	—	2	24	no	—
177	C₂³×C₄	D₄×C₁₃	1	24	yes	complemented
178	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
179	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
180	C₂³×C₄	C₂³×C₁₃	1	12	yes	—
181	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
182	#121	—	2	96	no	—
183	C₂×C₄⋊C₄	—	2	12	no	—
184	C₂×C₄⋊C₄	C₂³×C₁₃	1	12	yes	—
185	C₂×C₂²⋊C₄	—	2	24	no	—
186	C₂×C₂²⋊C₄	—	4	12	no	—
187	C₂×C₂²⋊C₄	C₂³×C₁₃	1	24	yes	—
188	C₂×C₂²⋊C₄	C₂³×C₁₃	1	24	yes	—
189	#95	C₂³×C₁₃	1	24	yes	—
190	C₂×C₁₃	—	4	12	no	—
191	C₂×C₁₃	—	8	4	no	—
192	C₂×C₁₃	#2615	1	8	yes	direct product factor
193	C₂×C₁₃	C₂²×#120	1	1	yes	characteristic
194	C₂×C₁₃	C₂×#535	1	3	yes	—
195	C₂×C₁₃	C₂×#537	1	3	yes	—
196	C₂⁴	—	2	12	no	—
197	C₂⁴	—	2	6	no	—
198	C₂⁴	—	2	12	no	—
199	C₂⁴	—	2	6	no	—
200	C₂⁴	—	2	12	no	—
201	C₂⁴	—	2	12	no	—
202	C₂⁴	—	2	4	no	—
203	C₂⁴	C₂×D₄×C₁₃	1	12	yes	—
204	C₂⁴	C₂⁴×C₁₃	1	4	yes	—
205	C₂⁴	C₂⁴×C₁₃	1	6	yes	—
206	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
207	#41	—	2	256	no	—
208	C₂×Q₈	—	2	32	no	—
209	C₂×Q₈	—	2	24	no	—
210	C₂×D₄	—	2	96	no	—
211	C₂×D₄	—	2	48	no	—
212	C₂×D₄	—	2	48	no	—
213	C₂×D₄	—	2	24	no	—
214	C₂×D₄	—	2	24	no	—
215	C₂×D₄	—	4	48	no	—
216	C₂×D₄	—	4	96	no	—
217	C₂×D₄	—	4	16	no	—
218	C₂²×C₄	—	2	12	no	—
219	C₂²×C₄	—	2	96	no	—
220	C₂²×C₄	—	2	48	no	—
221	C₂²×C₄	—	2	12	no	—
222	C₂²×C₄	—	2	24	no	—
223	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
225	C₂²×C₄	C₂×D₄×C₁₃	1	48	yes	complemented
226	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
227	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
228	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
229	C₄⋊C₄	—	2	24	no	—
230	C₂²⋊C₄	—	2	48	no	—
231	C₂²⋊C₄	—	4	24	no	—
232	C₁₃	C₂×#2615	1	1	yes	characteristic, direct product factor, Sylow
233	C₂³	—	2	24	no	—
234	C₂³	—	2	24	no	—
235	C₂³	—	2	24	no	—
236	C₂³	—	2	12	no	—
237	C₂³	—	2	6	no	—
238	C₂³	—	2	12	no	—
239	C₂³	—	2	6	no	—
240	C₂³	—	2	12	no	—
241	C₂³	—	2	12	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	4	24	no	—
244	C₂³	—	4	24	no	—
245	C₂³	—	4	8	no	—
246	C₂³	—	4	24	no	—
247	C₂³	—	4	24	no	—
248	C₂³	C₂²×D₄×C₁₃	1	12	yes	—
249	C₂³	C₂²×D₄×C₁₃	1	6	yes	—
250	C₂³	C₁₃×H(5,𝔽₂)	1	6	yes	—
251	C₂³	C₁₃×2₋¹⁺⁴	1	2	yes	—
252	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
253	Q₈	—	2	32	no	—
254	D₄	—	2	96	no	—
255	D₄	—	4	96	no	—
256	D₄	—	4	16	no	—
257	C₂×C₄	—	2	24	no	—
258	C₂×C₄	—	2	12	no	—
259	C₂×C₄	—	2	96	no	—
260	C₂×C₄	—	2	12	no	—
261	C₂×C₄	—	2	24	no	—
262	C₂×C₄	—	4	24	no	—
263	C₂×C₄	C₁₃×#120	1	16	yes	—
264	C₂×C₄	C₂²×D₄×C₁₃	1	12	yes	—
265	C₂×C₄	C₂²×D₄×C₁₃	1	12	yes	—
266	C₂²	—	2	12	no	—
267	C₂²	—	2	12	no	—
268	C₂²	—	2	12	no	—
269	C₂²	—	4	48	no	—
270	C₂²	—	4	12	no	—
271	C₂²	—	4	12	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	4	12	no	—
274	C₂²	—	4	12	no	—
275	C₂²	—	8	16	no	—
276	C₂²	C₂×C₁₃×#120	1	4	yes	—
277	C₂²	C₁₃×#535	1	12	yes	—
278	C₂²	C₁₃×#537	1	12	yes	—
279	C₂²	C₂³×D₄×C₁₃	1	3	yes	—
280	C₂²	C₂×C₁₃×H(5,𝔽₂)	1	3	yes	—
281	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
282	C₄	—	2	12	no	—
283	C₄	—	4	12	no	—
284	C₄	C₂×C₁₃×#120	1	8	yes	—
285	C₂	—	4	12	no	—
286	C₂	—	8	4	no	—
287	C₂	C₁₃×#2615	1	8	yes	direct product factor
288	C₂	C₂²×C₁₃×#120	1	1	yes	characteristic
289	C₂	C₂×C₁₃×#535	1	3	yes	—
290	C₂	C₂×C₁₃×#537	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×C₁₃×#41	C₂	semidirect
2. C₂²×C₁₃×#121	C₂	semidirect
3. C₁₃×#2615	C₂	direct (N × H)
4. C₂×C₁₃×#393	C₂	semidirect
5. C₂×C₁₃×#392	C₂	semidirect
6. C₂×C₁₃×#380	C₂	semidirect
8. C₂²×C₁₃×#41	C₂²	semidirect
15. C₂×C₁₃×#121	C₂²	semidirect
19. C₁₃×#393	C₂²	semidirect
20. C₁₃×#392	C₂²	semidirect
21. C₁₃×#380	C₂²	semidirect
38. C₂³×C₄×C₁₃	D₄	semidirect
51. C₂×#2615	C₁₃	direct (N × H)
81. C₂²×C₄×C₁₃	C₂×D₄	semidirect
88. C₂³×#41	C₂×C₁₃	semidirect
89. C₂²×#121	C₂×C₁₃	semidirect
90. #2615	C₂×C₁₃	direct (N × H)
91. C₂×#393	C₂×C₁₃	semidirect
92. C₂×#392	C₂×C₁₃	semidirect
93. C₂×#380	C₂×C₁₃	semidirect
128. C₂²×#41	C₂²×C₁₃	semidirect
135. C₂×#121	C₂²×C₁₃	semidirect
139. #393	C₂²×C₁₃	semidirect
140. #392	C₂²×C₁₃	semidirect
141. #380	C₂²×C₁₃	semidirect
177. C₂³×C₄	D₄×C₁₃	semidirect
192. C₂×C₁₃	#2615	direct (N × H)
225. C₂²×C₄	C₂×D₄×C₁₃	semidirect
232. C₁₃	C₂×#2615	direct (N × H)
287. C₂	C₁₃×#2615	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₁₃×#41	2
C₂²×C₁₃×#121	2
C₁₃×#2615	2
C₂×C₁₃×#393	2
C₂×C₁₃×#392	2
C₂×C₁₃×#380	2
C₂×#2615	13

Subgroups — C2×C13×#2615 #15720944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2615 #15720944